Function Machines

Put the input into the machine and do what the box says.

Do the first box, then put your answer into the second box.

Work through the boxes in order, from left to right.

Start at the output and work backwards, doing the opposite of each box.

For the output, work forwards. For the input, work backwards using opposite operations.

Do the opposite of dividing by 6.

1. The opposite of divide by 6 is multiply by 6: 4.5\times6=27.
2. Check: 27\div6=4.5. The input is 27.

Start at the output and undo the last box first.

1. The opposite of multiply by 4 is divide by 4: 120\div4=30.
2. The opposite of add 7 is subtract 7: 30-7=23.
3. Check: 23+7=30 and 30\times4=120. The input is 23.

Work backwards: undo the subtracting first, then undo the dividing.

1. The opposite of subtract 8 is add 8: 7+8=15.
2. The opposite of divide by 3 is multiply by 3: 15\times3=45.
3. Check: 45\div3=15 and 15-8=7. The input is 45.

A negative number multiplied by a positive number gives a negative answer.

1. First box: -5\times4=-20.
2. Second box: -20+9=-11.
3. The output is -11.

10 divided by 4 is not a whole number. Write it as a decimal.

1. First box: 10\div4=2.5.
2. Second box: 2.5+1.6=4.1.
3. The output is 4.1.

Put 7 through the first box. What do you need to add to get 41?

1. First box: 7\times5=35.
2. The second box must change 35 into 41: 41-35=6.
3. The missing number is 6, so the box is +\,6.

Work backwards from 69 through the second box first.

1. Undo add 6: 69-6=63.
2. So 9 times the missing number is 63, and 63\div9=7.
3. The missing number is 7, so the box is \times\,7.

Put 40 through the first box. What do you divide by to get 9?

1. First box: 40-4=36.
2. 36 divided by the missing number is 9, and 36\div9=4.
3. The missing number is 4, so the box is \div\,4.

Work out the output of each machine separately, then compare them.

1. Machine A: 4+3=7, then 7\times5=35.
2. Machine B: 4\times5=20, then 20+3=23.
3. 35-23=12. The output of machine A is 12 bigger.

When working backwards, undo the last box first.

Part (a)

1. The number goes through multiply by 2 first and add 5 last.
2. Working backwards, the add 5 must be undone first. Ravi undid the boxes in the wrong order.

Part (b)

1. Undo add 5: 21-5=16.
2. Undo multiply by 2: 16\div2=8.
3. Check: 8\times2=16 and 16+5=21. The input is 8.

Call the input n and follow it through the machine.

Part (a)

1. First box: n\times4=4n.
2. Second box: 4n-15.
3. The output is 4n-15.

Part (b)

1. The output equals the input, so 4n-15=n.
2. Subtract n from both sides: 3n-15=0, so 3n=15.
3. Divide by 3: n=5.
4. Check: 5\times4=20 and 20-15=5. The number is 5.

Call the input n. Write the output in terms of n and make it equal to 2n.

1. Let the input be n. The output is 3(n-4)=3n-12.
2. The output is twice the input: 3n-12=2n.
3. Subtract 2n from both sides: n-12=0, so n=12.
4. Check: 12-4=8 and 8\times3=24, which is twice 12. The input is 12.

Compare the two inputs and the two outputs.

Part (a)

1. The input goes up from 3 to 5, which is 2 more.
2. The output goes up from 17 to 27, which is 10 more.
3. Each 1 added to the input adds 10\div2=5 to the output, so the machine multiplies by 5.

Part (b)

1. 3\times5=15 and the output is 17, so the second box adds 17-15=2.
2. Check with the other input: 5\times5+2=27. The missing number is 2.

Call the input n. Write each output in terms of n and make them equal.

1. Let the input be n. Machine A gives 3n+10. Machine B gives 5n-4.
2. The outputs are the same: 3n+10=5n-4.
3. Subtract 3n from both sides: 10=2n-4. Add 4: 14=2n, so n=7.
4. Check: 3\times7+10=31 and 5\times7-4=31. The number is 7.

Work backwards through all three boxes, starting with the last one.

1. Undo multiply by 6: -6\div6=-1.
2. Undo subtract 3: -1+3=2.
3. Undo divide by 4: 2\times4=8.
4. Check: 8\div4=2, 2-3=-1 and -1\times6=-6. The input is 8.